Suppose

$$\begin{array}{rclcl} (t + 3) y_1^{\,\prime} & = & 8t y_1 + 9 y_2, && y_1(1) = 0, \\ (t - 4) y_2^{\,\prime} & = & 6 y_1 + 7t y_2, && y_2(1) = 2. \end{array}$$

1. This system of linear differential equations can be put in the form $\boldsymbol{\vec{y}^{\,\prime}} = P(t) \boldsymbol{\vec{y}} + \boldsymbol{\vec{g}}(t)$. Determine $P(t)$ and $\boldsymbol{\vec{g}}(t)$.

   $P(t) =$

   $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$ $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$

   $\boldsymbol{\vec{g}}(t) =$

   $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$ $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$

   
   
   
2. Is the system homogeneous or nonhomogeneous? Choose homogeneous nonhomogeneous
   
   
3. Find the largest interval $a < t < b$ such that a unique solution of the initial value problem is guaranteed to exist.
   
   Interval: help (inequalities)